§28.20 Definitions and Basic Properties

When $z$ is replaced by $\pm \mathrm{i}z$, (28.2.1) becomes the modified Mathieu’s equation:

28.20.1		$$w^{\prime \prime }-\left(a-2q\cosh \left(2z\right)\right)w=0,$$
ⓘ Symbols: $\cosh z$: hyperbolic cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution A&S Ref: 20.1.2 (in slightly different form) 20.8.6 Referenced by: §28.20(iii), §28.32(i), 1st item, 2nd item, item (b), §28.8(iv), §28.8(iv) Permalink: http://dlmf.nist.gov/28.20.E1 Encodings: TeX, pMML, png See also: Annotations for §28.20(i), §28.20 and Ch.28

with its algebraic form

28.20.2		$$({\zeta }^2-1)w^{\prime \prime }+\zeta w^{\prime }+\left(4q{\zeta }^2-2q-a\right)w=0,$$
$\zeta =\cosh z$.
ⓘ Symbols: $\cosh z$: hyperbolic cosine function, $q=h^2$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution Referenced by: §28.20(iii) Permalink: http://dlmf.nist.gov/28.20.E2 Encodings: TeX, pMML, png See also: Annotations for §28.20(i), §28.20 and Ch.28

28.20.3		${\mathrm{Ce}}_{\nu }(z,q)$	$={\mathrm{ce}}_{\nu }(\pm \mathrm{i}z,q),$
$\nu \ne -1,-2,\mathrm{\dots }$,
ⓘ Defines: ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function Symbols: ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter A&S Ref: 20.6.1 (in slightly different form) 20.6.2 (in slightly different form) Permalink: http://dlmf.nist.gov/28.20.E3 Encodings: TeX, pMML, png See also: Annotations for §28.20(ii), §28.20 and Ch.28
28.20.4		${\mathrm{Se}}_{\nu }(z,q)$	$=\mp \mathrm{i}{\mathrm{se}}_{\nu }(\pm \mathrm{i}z,q),$
$\nu \ne 0,-1,\mathrm{\dots }$,
ⓘ Defines: ${\mathrm{Se}}_{\nu }(z,q)$: modified Mathieu function Symbols: ${\mathrm{se}}_n(z,q)$: Mathieu function, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter A&S Ref: 20.6.1 (in slightly different form) 20.6.2 (in slightly different form) Permalink: http://dlmf.nist.gov/28.20.E4 Encodings: TeX, pMML, png See also: Annotations for §28.20(ii), §28.20 and Ch.28
28.20.5		${\mathrm{Me}}_{\nu }(z,q)$	$={\mathrm{me}}_{\nu }(-\mathrm{i}z,q),$
ⓘ Defines: ${\mathrm{Me}}_{\nu }(z,q)$: modified Mathieu function Symbols: ${\mathrm{me}}_n(z,q)$: Mathieu function, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter A&S Ref: 20.6.1 (in slightly different form) 20.6.2 (in slightly different form) Permalink: http://dlmf.nist.gov/28.20.E5 Encodings: TeX, pMML, png See also: Annotations for §28.20(ii), §28.20 and Ch.28
28.20.6		${\mathrm{Fe}}_n(z,q)$	$=\mp \mathrm{i}{\mathrm{fe}}_n(\pm \mathrm{i}z,q),$
$n=0,1,\mathrm{\dots }$,
ⓘ Defines: ${\mathrm{Fe}}_n(z,q)$: modified Mathieu function Symbols: ${\mathrm{fe}}_n(z,q)$: second solution, Mathieu’s equation, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $n$: integer and $z$: complex variable A&S Ref: 20.6.1 (in slightly different form) 20.6.2 (in slightly different form) Permalink: http://dlmf.nist.gov/28.20.E6 Encodings: TeX, pMML, png See also: Annotations for §28.20(ii), §28.20 and Ch.28
28.20.7		${\mathrm{Ge}}_n(z,q)$	$={\mathrm{ge}}_n(\pm \mathrm{i}z,q),$
$n=1,2,\mathrm{\dots }$.
ⓘ Defines: ${\mathrm{Ge}}_n(z,q)$: modified Mathieu function Symbols: ${\mathrm{ge}}_n(z,q)$: second solution, Mathieu’s equation, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $n$: integer and $z$: complex variable A&S Ref: 20.6.1 (in slightly different form) 20.6.2 (in slightly different form) Permalink: http://dlmf.nist.gov/28.20.E7 Encodings: TeX, pMML, png See also: Annotations for §28.20(ii), §28.20 and Ch.28

Assume first that $\nu$ is real, $q$ is positive, and $a={\lambda }_{\nu }\left(q\right)$; see §28.12(i). Write

28.20.8		$$h=\sqrt{q}(>0).$$
ⓘ Symbols: $q=h^2$: parameter and $h$: parameter Permalink: http://dlmf.nist.gov/28.20.E8 Encodings: TeX, pMML, png See also: Annotations for §28.20(iii), §28.20 and Ch.28

Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to ${\zeta }^{1/2}{\mathrm{e}}^{\pm 2\mathrm{i}h\zeta }$ as $\zeta \to \mathrm{\infty }$ in the respective sectors $|\mathrm{ph}\left(\mp \mathrm{i}\zeta \right)|\le \frac{3}{2}\pi -\delta$, $\delta$ being an arbitrary small positive constant. It follows that (28.20.1) has independent and unique solutions ${\mathrm{M}}_{\nu }^{(3)}(z,h)$, ${\mathrm{M}}_{\nu }^{(4)}(z,h)$ such that

28.20.9		$${\mathrm{M}}_{\nu }^{(3)}(z,h)=H_{\nu }^{(1)}\left(2h\cosh z\right)\left(1+O\left(\mathrm{sech}z\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/28.20.E9 Encodings: TeX, pMML, png See also: Annotations for §28.20(iii), §28.20 and Ch.28

as $\mathrm{\Re }z\to +\mathrm{\infty }$ with $-\pi +\delta \le \mathrm{\Im }z\le 2\pi -\delta$, and

28.20.10		$${\mathrm{M}}_{\nu }^{(4)}(z,h)=H_{\nu }^{(2)}\left(2h\cosh z\right)\left(1+O\left(\mathrm{sech}z\right)\right),$$
ⓘ Symbols: $O\left(x\right)$: order not exceeding, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/28.20.E10 Encodings: TeX, pMML, png See also: Annotations for §28.20(iii), §28.20 and Ch.28

as $\mathrm{\Re }z\to +\mathrm{\infty }$ with $-2\pi +\delta \le \mathrm{\Im }z\le \pi -\delta$. See §10.2(ii) for the notation. In addition, there are unique solutions ${\mathrm{M}}_{\nu }^{(1)}(z,h)$, ${\mathrm{M}}_{\nu }^{(2)}(z,h)$ that are real when $z$ is real and have the properties

28.20.11		$${\mathrm{M}}_{\nu }^{(1)}(z,h)=J_{\nu }\left(2h\cosh z\right)+{\mathrm{e}}^{|\mathrm{\Im }\left(2h\cosh z\right)|}O\left({\left(\mathrm{sech}z\right)}^{3/2}\right),$$
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\mathrm{\Im }$: imaginary part, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter Referenced by: §28.33(ii) Permalink: http://dlmf.nist.gov/28.20.E11 Encodings: TeX, pMML, png See also: Annotations for §28.20(iii), §28.20 and Ch.28

28.20.12		$${\mathrm{M}}_{\nu }^{(2)}(z,h)=Y_{\nu }\left(2h\cosh z\right)+{\mathrm{e}}^{|\mathrm{\Im }\left(2h\cosh z\right)|}O\left({(\mathrm{sech}z)}^{3/2}\right),$$
ⓘ Symbols: $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\mathrm{\Im }$: imaginary part, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter Referenced by: §28.33(ii) Permalink: http://dlmf.nist.gov/28.20.E12 Encodings: TeX, pMML, png See also: Annotations for §28.20(iii), §28.20 and Ch.28

as $\mathrm{\Re }z\to +\mathrm{\infty }$ with $|\mathrm{\Im }z|\le \pi -\delta$.

For other values of $z$, $h$, and $\nu$ the functions ${\mathrm{M}}_{\nu }^{(j)}(z,h)$, $j=1,2,3,4$, are determined by analytic continuation. Furthermore,

28.20.13		${\mathrm{M}}_{\nu }^{(3)}(z,h)$	$={\mathrm{M}}_{\nu }^{(1)}(z,h)+\mathrm{i}{\mathrm{M}}_{\nu }^{(2)}(z,h),$
ⓘ Symbols: $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter A&S Ref: 20.6.16 (only for integer $\nu$) Referenced by: §28.22(ii) Permalink: http://dlmf.nist.gov/28.20.E13 Encodings: TeX, pMML, png See also: Annotations for §28.20(iii), §28.20 and Ch.28
28.20.14		${\mathrm{M}}_{\nu }^{(4)}(z,h)$	$={\mathrm{M}}_{\nu }^{(1)}(z,h)-\mathrm{i}{\mathrm{M}}_{\nu }^{(2)}(z,h).$
ⓘ Symbols: $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter Referenced by: §28.22(ii) Permalink: http://dlmf.nist.gov/28.20.E14 Encodings: TeX, pMML, png See also: Annotations for §28.20(iii), §28.20 and Ch.28

For $j=1,2,3,4$,

28.20.15		${\mathrm{Mc}}_n^{(j)}(z,h)$	$={\mathrm{M}}_n^{(j)}(z,h),$
$n=0,1,\mathrm{\dots }$,
ⓘ Defines: ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function Symbols: ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer and $z$: complex variable Referenced by: §28.20(vii) Permalink: http://dlmf.nist.gov/28.20.E15 Encodings: TeX, pMML, png See also: Annotations for §28.20(iv), §28.20 and Ch.28
28.20.16		${\mathrm{Ms}}_n^{(j)}(z,h)$	$={(-1)}^n{\mathrm{M}}_{-n}^{(j)}(z,h),$
$n=1,2,\mathrm{\dots }$.
ⓘ Defines: ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function Symbols: ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer and $z$: complex variable Referenced by: §28.20(vii) Permalink: http://dlmf.nist.gov/28.20.E16 Encodings: TeX, pMML, png See also: Annotations for §28.20(iv), §28.20 and Ch.28

28.20.17		${\mathrm{Ie}}_n(z,h)$	$={\mathrm{i}}^{-n}{\mathrm{Mc}}_n^{(1)}(z,\mathrm{i}h),$
ⓘ Defines: ${\mathrm{Ie}}_n(z,h)$: modified Mathieu function Symbols: $\mathrm{i}$: imaginary unit, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer and $z$: complex variable A&S Ref: 20.8.10 (in slightly different form) Permalink: http://dlmf.nist.gov/28.20.E17 Encodings: TeX, pMML, png See also: Annotations for §28.20(v), §28.20 and Ch.28
28.20.18		${\mathrm{Io}}_n(z,h)$	$={\mathrm{i}}^{-n}{\mathrm{Ms}}_n^{(1)}(z,\mathrm{i}h),$
ⓘ Defines: ${\mathrm{Io}}_n(z,h)$: modified Mathieu function Symbols: $\mathrm{i}$: imaginary unit, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer and $z$: complex variable A&S Ref: 20.8.10 (in slightly different form) Permalink: http://dlmf.nist.gov/28.20.E18 Encodings: TeX, pMML, png See also: Annotations for §28.20(v), §28.20 and Ch.28

28.20.19		${\mathrm{Ke}}_{2m}(z,h)$	$={(-1)}^m\frac{1}{2}\pi \mathrm{i}{\mathrm{Mc}}_{2m}^{(3)}(z,\mathrm{i}h),$
		${\mathrm{Ke}}_{2m+1}(z,h)$	$={(-1)}^{m+1}\frac{1}{2}\pi {\mathrm{Mc}}_{2m+1}^{(3)}(z,\mathrm{i}h),$
ⓘ Defines: ${\mathrm{Ke}}_n(z,h)$: modified Mathieu function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $n$: integer and $z$: complex variable A&S Ref: 20.8.11 (in slightly different form) Permalink: http://dlmf.nist.gov/28.20.E19 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.20(v), §28.20 and Ch.28

28.20.20		${\mathrm{Ko}}_{2m}(z,h)$	$={(-1)}^m\frac{1}{2}\pi \mathrm{i}{\mathrm{Ms}}_{2m}^{(3)}(z,\mathrm{i}h),$
		${\mathrm{Ko}}_{2m+1}(z,h)$	$={(-1)}^{m+1}\frac{1}{2}\pi {\mathrm{Ms}}_{2m+1}^{(3)}(z,\mathrm{i}h).$
ⓘ Defines: ${\mathrm{Ko}}_n(z,h)$: modified Mathieu function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $n$: integer and $z$: complex variable Permalink: http://dlmf.nist.gov/28.20.E20 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.20(v), §28.20 and Ch.28

28.20.21		$𝒲\left\{{\mathrm{M}}_{\nu }^{(1)},{\mathrm{M}}_{\nu }^{(2)}\right\}$	$=-𝒲\left\{{\mathrm{M}}_{\nu }^{(2)},{\mathrm{M}}_{\nu }^{(3)}\right\}=-𝒲\left\{{\mathrm{M}}_{\nu }^{(2)},{\mathrm{M}}_{\nu }^{(4)}\right\}=2/\pi ,$
		$𝒲\left\{{\mathrm{M}}_{\nu }^{(1)},{\mathrm{M}}_{\nu }^{(3)}\right\}$	$=-𝒲\left\{{\mathrm{M}}_{\nu }^{(1)},{\mathrm{M}}_{\nu }^{(4)}\right\}=-\frac{1}{2}𝒲\left\{{\mathrm{M}}_{\nu }^{(3)},{\mathrm{M}}_{\nu }^{(4)}\right\}=2\mathrm{i}/\pi .$
ⓘ Symbols: $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/28.20.E21 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.20(vi), §28.20 and Ch.28

28.20.22		$${\mathrm{M}}_{\nu }^{(j)}(z\pm \frac{1}{2}\pi \mathrm{i},h)={\mathrm{M}}_{\nu }^{(j)}(z,\pm \mathrm{i}h),$$
$\nu \notin \mathbb{Z}$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $\notin$: not an element of, $h$: parameter, $j$: integer, $z$: complex variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/28.20.E22 Encodings: TeX, pMML, png See also: Annotations for §28.20(vii), §28.20 and Ch.28

For $n=0,1,2,\mathrm{\dots }$,

28.20.23		${\mathrm{Mc}}_{2n}^{(j)}(z\pm \frac{1}{2}\pi \mathrm{i},h)$	$={\mathrm{Mc}}_{2n}^{(j)}(z,\pm \mathrm{i}h),$
		${\mathrm{Ms}}_{2n+1}^{(j)}(z\pm \frac{1}{2}\pi \mathrm{i},h)$	$={\mathrm{Mc}}_{2n+1}^{(j)}(z,\pm \mathrm{i}h),$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer, $j$: integer and $z$: complex variable A&S Ref: 20.8.7 (in slightly different form) Permalink: http://dlmf.nist.gov/28.20.E23 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.20(vii), §28.20 and Ch.28

28.20.24		${\mathrm{Mc}}_{2n+1}^{(j)}(z\pm \frac{1}{2}\pi \mathrm{i},h)$	$={\mathrm{Ms}}_{2n+1}^{(j)}(z,\pm \mathrm{i}h),$
		${\mathrm{Ms}}_{2n+2}^{(j)}(z\pm \frac{1}{2}\pi \mathrm{i},h)$	$={\mathrm{Ms}}_{2n+2}^{(j)}(z,\pm \mathrm{i}h).$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer, $j$: integer and $z$: complex variable A&S Ref: 20.8.7 (in slightly different form) Permalink: http://dlmf.nist.gov/28.20.E24 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.20(vii), §28.20 and Ch.28

For $s\in \mathbb{Z}$,

28.20.25		${\mathrm{M}}_{\nu }^{(1)}(z+s\pi \mathrm{i},h)$	$={\mathrm{e}}^{\mathrm{i}s\pi \nu }{\mathrm{M}}_{\nu }^{(1)}(z,h),$
		${\mathrm{M}}_{\nu }^{(2)}(z+s\pi \mathrm{i},h)$	$={\mathrm{e}}^{-\mathrm{i}s\pi \nu }{\mathrm{M}}_{\nu }^{(2)}(z,h)+2\mathrm{i}\cot \left(\pi \nu \right)\sin \left(s\pi \nu \right){\mathrm{M}}_{\nu }^{(1)}(z,h),$
		${\mathrm{M}}_{\nu }^{(3)}(z+s\pi \mathrm{i},h)$	$=-{\displaystyle \frac{\sin \left((s-1)\pi \nu \right)}{\sin \left(\pi \nu \right)}}{\mathrm{M}}_{\nu }^{(3)}(z,h)-{\mathrm{e}}^{-\mathrm{i}\pi \nu }{\displaystyle \frac{\sin \left(s\pi \nu \right)}{\sin \left(\pi \nu \right)}}{\mathrm{M}}_{\nu }^{(4)}(z,h),$
		${\mathrm{M}}_{\nu }^{(4)}(z+s\pi \mathrm{i},h)$	$={\mathrm{e}}^{\mathrm{i}\pi \nu }{\displaystyle \frac{\sin \left(s\pi \nu \right)}{\sin \left(\pi \nu \right)}}{\mathrm{M}}_{\nu }^{(3)}(z,h)+{\displaystyle \frac{\sin \left((s+1)\pi \nu \right)}{\sin \left(\pi \nu \right)}}{\mathrm{M}}_{\nu }^{(4)}(z,h).$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $\sin z$: sine function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter Referenced by: §28.20(vii) Permalink: http://dlmf.nist.gov/28.20.E25 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §28.20(vii), §28.20 and Ch.28

When $\nu$ is an integer the right-hand sides of (28.20.25) are replaced by the their limiting values. And for the corresponding identities for the radial functions use (28.20.15) and (28.20.16).